Statistics

Data Collection, Classification, and Frequency Tables

A statistical question anticipates variability in answers and specifies a population, unlike a question with a single fixed answer.

• The population is the entire group studied; a sample is a subset chosen to represent it.
• Avoid bias by choosing a representative sample, asking neutral questions, and sampling randomly across groups.
• Categorical data consists of descriptive labels; numerical data consists of numbers, either discrete (counted) or continuous (measured).
• A frequency table lists each value or class interval alongside how often it occurs.
• Use a tally to count frequency; $\sum f$ is the total number of data values.
• Grouped frequency tables use class intervals (such as $0 \le t < 10$) for continuous data.

Graphical Displays and Back-to-Back Stem-and-Leaf Plots

Different graph types suit different variables:

• Bar charts: Show categorical or discrete data using equal-width bars separated by gaps.
• Histograms: Display continuous grouped data using equal intervals where neighbouring bars touch without gaps.
• Pie charts: Represent proportions of $360^\circ$, where $\text{Angle} = \frac{\text{Frequency}}{\text{Total Frequency}} \times 360^\circ$.
• Line plots: Stack dots or crosses above a number line to show frequencies in small datasets.

An ordered back-to-back stem-and-leaf plot compares two distributions using a shared central column of stems:

• Leaves on the right read normally from left to right, smallest closest to the stem.
• Leaves on the left read backwards: smallest digit closest to stem, largest furthest left.
• Write each leaf as a single digit, spaced evenly without commas.
• Always include a key showing how to read both sides, including units.

Measures of Centre, Spread, and Grouped Means

• Mode: The most frequent value. For grouped data, the modal class has the highest frequency.
• Median: The middle value of ordered data, or the average of the two middle values.
• Mean: Add the values and divide by their count, $n$: $\bar{x} = \frac{\sum x}{n}$, so $\text{Total} = \text{Mean} \times n$.
• Range: The spread of values: $\text{Range} = \text{highest value} - \text{lowest value}$.
• Estimated mean: Use mid-interval values $x = \frac{\text{lower limit} + \text{upper limit}}{2}$ when exact grouped values are unknown:

where $f$ is frequency and $x$ is the midpoint.

Outliers, Misleading Graphs, and Evaluating Investigations

• An outlier lies far from other values; it can pull the mean towards it and increase the range.
• The median is less affected by extreme outliers than the mean.
• To identify an outlier, state the value and explain that it lies far from the cluster.
• A bar chart whose vertical axis starts above zero can exaggerate differences. Check the scale.
• A representative sample can suggest patterns in the population, but conclusions remain uncertain.

Key terms

Statistical Question

A question that anticipates answers that vary and specifies a population and measurement of interest.

Population

The entire group of people or items being studied in an investigation.

Sample

A smaller group selected from a population to represent the whole.

Frequency Table

A table that lists values or class intervals alongside the count of how often each occurs.

Mid-Interval Value

The exact midpoint of a class interval, calculated as (lower limit + upper limit) / 2.

Range

The difference between the highest value and the lowest value in a dataset.

Outlier

A data point located far away from the rest of the values, which can distort the mean and range.

Confounding Variable

An unmeasured factor that may influence the results of a study and distort conclusions.